SDM Benchmark Study Part 3: Baseline Models

Overview

Inhomogeneous Poisson Process (IPP)

An Inhomogeneous Poisson Process (IPP) is a statistical model used for events that occur randomly over space or time. Unlike a regular Poisson process, the rate of event occurrence in an IPP can vary.

IPP is often used in presence-only problems to model the intensity of events across different locations or times. It’s defined as \(\lambda(x) = N * f(x)\), where \(\lambda(x)\) is the intensity function, \(N\) is the total number of events, and \(f(x)\) is the density function.

By fitting an IPP to presence-only data, we estimate the underlying intensity function, which tells us how the rate of event occurrence changes across different locations or times. For example, we might hypothesize that the presence of the Cedar Waxwing is influenced by factors like canopy cover, land cover type, and temperature. We use the presence-only data to fit the IPP model, estimating the parameters of \(\lambda(x)\) that best fit the observed data.

The intensity function \(\lambda(x)\) in an IPP is typically defined as a function of some parameters \(\theta\) and the environmental variables at location \(x\). For example, we might have For example, \(\lambda(x; \theta) = \exp(\theta_1 * canopy + \theta_2 * land\_cover + ...)\), where \(canopy\), \(land\_cover\), etc. are the environmental variables, and \(\theta_1\), \(\theta_2\), etc. are the parameters to be estimated.

The likelihood of the observed data given the parameters \(\theta\) is given by:

\(L(θ) = [∏ λ(x_i; θ)] * exp(-∫ λ(x; θ) dx)\), where the product is over all observed presence locations \(x_i\), and the integral is over all possible locations \(x\). The first part of this formula represents the probability of observing the species at the observed locations, and the second part represents the probability of not observing the species at any other location.

The goal of maximum likelihood estimation is to find the parameters \(θ\) that maximize this likelihood. This is typically done using numerical optimization methods, such as gradient descent or Newton’s method.

Once we have estimated the parameters \(θ\), we can use them to calculate the intensity function \(λ(x; θ)\) at any location \(x\). This gives us an estimate of the rate of species occurrence at that location, based on the environmental variables at that location.

IPP Assumptions

As with any parametric statistical model, one of the biggest constraints is that there are certain assumptions that must be met in order for the model to technically be “valid”.

• Independence: IPP assumes that the events occur independently in space or time, meaning the occurrence of an event at one location or time does not affect the occurrence of an event at another location or time.
• Inhomogeneity: IPP assumes that the intensity function can vary across space or time, as opposed to a homogeneous Poisson process, which assumes a constant rate of event occurrence.
• Known Intensity Function: IPP assumes that the form of the intensity function is known, although the parameters of the function need to be estimated from the data. This assumption can be violated if the true intensity function is not well-captured by the chosen form.
• Complete Spatial Coverage: IPP assumes that the entire study area has been surveyed and that presence data is available for all locations where the species is present. This assumption can be violated due to incomplete or biased sampling.

Maximum Entropy (MaxEnt)

Entropy is a measure of uncertainty, randomness, or chaos in a set of data. In other words, it quantifies the amount of unpredictability or surprise in possible outcomes.

Maximum Entropy (MaxEnt) is a method that selects the most spread-out probability distribution fitting our known data. It’s useful in presence-only problems as it minimizes bias in estimating event occurrence based on observed data. “It agrees with everything that is known, but carefully avoids assuming anything that is not known” (Jaynes, 1990).

MaxEnt Assumptions

• Incomplete Information: MaxEnt assumes that we do not have complete information about the system we are modeling. It uses the available information (presence data and environmental variables) to make the least biased estimate of the probability distribution.
• Feature Independence: MaxEnt assumes that the features (environmental variables) are independent. In reality, this assumption is often violated as environmental variables can be correlated.
• Linear Response: MaxEnt assumes a linear response in the log odds of presence with respect to the environmental variables. This assumption can be relaxed by including interaction terms and quadratic terms in the model.
  • The “linear response” assumption in MaxEnt refers to the idea that the log odds of presence (i.e., the natural logarithm of the odds of a species being present versus absent) is a linear function of the environmental variables. In mathematical terms, if \(P\) is the probability of presence, the log odds of presence is \(log(P/(1-P))\). The linear response assumption means that this quantity is modeled as a linear function of the environmental variables. For example, if we have two environmental variables \(x_1\) and \(x_2\), the linear response model would be: \(log(P/(1-P)) = β_0 + β_1x_1 + β_2x_2\), where \(β_0\), \(β_1\), and \(β_2\) are parameters to be estimated from the data.
  • This is a simplifying assumption that makes the model easier to estimate and interpret. However, it may not always be realistic. For example, the relationship between the probability of presence and the environmental variables might be nonlinear, or there might be interactions between the environmental variables.
  • To accommodate these possibilities, we can include interaction terms and quadratic terms in the model. An interaction term (e.g., \(x_1*x_2\)) allows the effect of one variable to depend on the value of another variable. A quadratic term (e.g., \(x_1^2\) or \(x_2^2\)) allows for a nonlinear relationship between the probability of presence and the environmental variable.
  • By including these terms in the model, we can relax the linear response assumption and potentially achieve a better fit to the data. However, this also makes the model more complex and potentially harder to interpret.
• Sample Representativeness: MaxEnt assumes that the presence data is representative of the species’ distribution. This means that the locations where the species is observed are a random sample from the species’ actual distribution.

Loading the Data

Load the Tabular Data

# Load the dataset saved in part 2 of the study
df <- readRDS("artifacts/final_data/final_data.rds")

# Define some global variables that will be referenced throughout the modeling 
states <- sort(unique(df$state))
species <- sort(unique(df$common.name))

# Convert to data.table
df %>% setDT()

# View output
df %>% as_tibble()

Load Rasters

# Load "Feature Engineered" rasters and original rasters into a 
# single multi-layer raster by state
r.list <- set_names(states) %>%
  purrr::map(~rast(c(paste0("data/final_rasters/", .x, ".tif"),
                     file.path("artifacts/feature_engineered_final", 
                               paste0(.x, ".tif")))))

# Create plots of NC Rasters as an example
nc.rast.plts <- purrr::map(names(r.list$NC), function(r.name) {
  r.df <- terra::as.data.frame(r.list$NC[[r.name]], xy=T)
  p <- ggplot(r.df, aes(x=x, y=y, fill=!!sym(r.name))) +
    geom_raster() +
    coord_cartesian() 
    if (r.name != "NLCD_Land") p <- p + scale_fill_viridis_c()
  p + labs(title=r.name) + theme(legend.position="none")
}) %>%
  ggarrange(plotlist=., 
            ncol=4, nrow=ceiling(length(names(r.list$NC)) / 4)) +
  ggtitle("All Raster Layers for NC")

# View the plots of all of the NC rasters
print(nc.rast.plts)